Given a triangular base pyramid with the following dimensions
[tex]\begin{gathered} For\text{ the lateral sides} \\ h=12m \\ b=8m \\ \text{For the base} \\ h=6.9m \\ b=8m \end{gathered}[/tex]
To find the total surface area, TSA, of the triangular base pyramid, the formula is
[tex]\text{TSA}=\text{base area}+\frac{1}{2}(Perimeter\times slant\text{ height)}[/tex]
The base area of the triangular base pyramid is
[tex]\text{Base area}=\frac{1}{2}bh=\frac{1}{2}\times8\times6.9=27.6m^2[/tex]
For the other part of the total surface area
[tex]\begin{gathered} \frac{1}{2}(Perimeter\times slant\text{ height)}=\frac{1}{2}((8+8+8)\times12) \\ =\frac{1}{2}(24\times12)=\frac{1}{2}\times288=144m^2 \end{gathered}[/tex]
The total surface area of the triangular base pyramid is
[tex]\text{TSA}=\text{base area}+\frac{1}{2}(Perimeter\times slant\text{ height)}=27.6+144=171.6m^2[/tex]
Hence, the total surface area of the triangular base pyramid is 171.6m²
To find the lateral area of a triangular base pyramid, the formula is
[tex]\begin{gathered} Lateral\text{ area}=\frac{1}{2}(Perimeter\times slant\text{ height)} \\ =\frac{1}{2}((8+8+8)\times12) \\ \frac{1}{2}(24\times12)=\frac{1}{2}(24\times12)=\frac{1}{2}(288)=144m^2 \\ Lateral\text{ area}=144m^2 \end{gathered}[/tex]
Hence, the lateral area of the triangular base pyramid is 144m²
